222 XIII INTEGRATION
function u e Jf(X; L) (resp. v e Jf (Y; M)), we have \k(u)\ g aL \\u\\ (resp.
\H(v)\ <* bM \\v\\). Hence, for each function h e Jf(X x Y; L x M), we have
II'
h(x,
for all y e Y, and by virtue of (13.21.1.3) and the definition of v(K),
Since every compact subset of X x Y is contained in the product of its projec-
tions on X and on Y, the proof of (13.21.1) is complete.
The number v(/z) = fi((h(x9 y) dk(x)\ is also denoted by
[
Since we can clearly interchange the roles of X and Y, it follows that
(13.21.2) !h(x, y) dv(x, y) = t dl(x) th(x9 y)
= (
for all h e jf C(X x Y).
By reason of this formula we write f (h dl dj* or |T h(x, y) dh(x) d/j,(y)
instead of \h(x, y) dv(x, y), and we may also interchange /I and \JL in these
notations. The measure v is called the product of 1 and n, and is denoted by
I®/*. It is clear that the mapping (A, /4)i-»A® \JL of MC(X) x MC(Y) into
MC(X x Y) is bilinear: in other words, we have
and
= fl(A ® ILL)
for any scalar a.
Furthermore, if A and fj, are m*7 (resp. positive) measures, then so is
(1 3.21 .3) Let 1, ILL be two positive measures on X, Y respectively, and v = 1 ® j
their product. For each function h e «/(X x Y), the function
x^\tion h e ^TC(X x Y), the